\section{Orbital maneuvers}

\begin{frame}{\thesection.\ \insertsection}
\begin{itemize}\setlength{\itemsep}{16pt}
    \item The communication satellites typically fly in Geostationary Earth Orbit (GEO), about 36000km above the Earth.
    \item When a satellite is injected into orbit using some launch vehicle, the point of injection is roughly above the launch site.
    \item It is very difficult to launch a satellite directly into GEO, in an equatorial orbit (unless the launch site is very close to the equator).
\end{itemize}
\begin{center}\includegraphics[scale=0.4]{fig_4_p34.jpg}\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection}
\begin{itemize}
\item The classic procedure is an injection into a highly eccentric orbit called GTO
    (Geostationary Transfer Orbit) followed by a number of satellite maneuvers aimed at reaching the final GEO.
    \begin{center}\includegraphics{fig_4_p35.pdf}\end{center}
\item When a satellite is launched, we typically want a specific orbit depending on the mission objectives.
    \begin{itemize}
    \item[\mysquare] This orbit may or may not be achievable directly from launch,
        given the launch site or type of launch vehicle.
    \end{itemize}
\item Therefore, orbital maneuvers are often required.
\end{itemize}
\end{frame}

\begin{frame}{\thesection.\ \insertsection}
    Maneuvers are performed using onboard thrusters, typically (although not always) in a sequence of short duration bursts.
    \begin{center}\includegraphics[scale=0.4]{fig_4_1.jpg}\end{center}
    \begin{center}\textcolor{blue}{Figure \arabic{section}.1:} Apogee engine\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection}
For our purposes, \textcolor{blue}{we will assume that these bursts are impulsive}.
\begin{block}{Impusive burst}
\begin{itemize}
    \item causing an instantaneous change to the spacecraft velocity vector \(\vec{v}\)
    \vspace{-6pt}
    \[ \triangle \vec{v} = \vec{v}_2 - \vec{v}_1 \]
    \vspace{-24pt}
    \item without affecting the position \(\vec{_{}r}\)
    \vspace{-6pt}
    \[ \triangle \vec{_{}r} = 0 \]
\end{itemize}
\end{block}
\begin{center}\includegraphics{fig_4_2.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.2:} Single impulse maneuver\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simple impulsive maneuvers}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The simplest type of maneuver requires only a single impulse.
\begin{center}\includegraphics{fig_4_2.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.3:} Single impulse maneuver\end{center}
An important parameter of the maneuver is the magnitude of the velocity change
\[ \Delta v = |\Delta \vec{v}| \]
which is \textcolor{blue}{a measure of the fuel consumption}.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Coplanar maneuvers}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Let us assume that the initial and final orbits lie \textcolor{blue}{in the same plane}. \\
We will restrict ourselves to \textcolor{blue}{tangential velocity changes}.
\begin{block}{Tangential velocity changes}
\begin{itemize}
    \item only change the velocity magnitude
    \item but not change the velocity direction
\end{itemize}
\end{block}
\begin{columns}
\column{0.33\textwidth}
    \begin{center}\includegraphics{fig_4_3.pdf}\end{center}
    \textcolor{blue}{Figure \arabic{section}.3:} Circular to elliptical transfer
\column{0.33\textwidth}
    \begin{center}\includegraphics{fig_4_4.pdf}\end{center}
    \textcolor{blue}{Figure \arabic{section}.4:} Elliptical to circular transfer
\column{0.33\textwidth}
    \begin{center}\includegraphics{fig_4_5.pdf}\end{center}
    \textcolor{blue}{Figure \arabic{section}.5:} Hohmann transfer
\end{columns}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
1. Circular to elliptical transfer
\begin{itemize}
    \item For circular orbits, \( \vec{_{}r} \perp \vec{v} \).
    \item For elliptical orbits, only at periapsis and apoapsis, \( \vec{_{}r} \perp \vec{v} \).
\end{itemize}
This means that any tangential velocity change at any point on a circular orbit results in that point becoming either periapsis or apoapsis of the new orbit.
\begin{center}\includegraphics{fig_4_6.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.6:} Circular to elliptical transfer\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
According to the \textit{vis-viva equation}
\[ v = \sqrt{\mu \left( \frac{2}{r} - \frac{1}{a} \right)} \]
We have
\[ \Delta v = v_2 - v_1 = \sqrt{\mu \left( \frac{2}{r_1} - \frac{1}{a} \right)} - \sqrt{\frac{\mu}{r_1}} \]
\vspace{-12pt}
\begin{center}\includegraphics{fig_4_6.pdf}\end{center}
\vspace{-12pt}
\begin{columns}
\column{0.45\textwidth}
If \( a > r_1 \), we have \( v_2 > v_1 \).
\column{0.45\textwidth}
If \( a < r_1 \), we have \( v_2 < v_1 \).
\end{columns}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
2. Elliptical to circular transfer
\begin{itemize}
    \item For circular orbits, \( \vec{_{}r} \perp \vec{v} \).
    \item For elliptical orbits, only at periapsis and apoapsis, \( \vec{_{}r} \perp \vec{v} \).
\end{itemize}
\begin{center}\includegraphics{fig_4_7.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.7:} Elliptical to circular transfer\end{center}
If we wish to circularize an elliptical orbit using a tangential transfer, we can only do so at periapsis or apoapsis.
\[ \Delta v = v_1 - v_2 = \sqrt{\frac{\mu}{r_2}} - \sqrt{\mu \left( \frac{2}{r_2} - \frac{1}{a} \right)} \]
\end{frame}

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\frametitle{}
3. Hohmann transfer
\vfill
Consider a transfer between two circular orbits of radius \( r_1 \) and \( r_2 \).
\begin{columns}
\column{0.45\textwidth}
    \begin{center}\includegraphics{fig_4_8.pdf}\end{center}
    \begin{center}\textcolor{blue}{Figure \arabic{section}.8:} Hohmann transfer\end{center}
\column{0.45\textwidth}
    \begin{itemize}
        \item These orbits do not have a point in common.
        \item Therefore, a single impulse transfer is not possible.
    \end{itemize}
\end{columns}

\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\vspace{-8pt}
\begin{columns}
\column{0.65\textwidth}
\begin{block}{Walter Hohmann proposed a double impulse transfer in 1925.}
\begin{itemize}
    \item It consists of two tangential maneuvers: a circular to elliptical transfer followed by an elliptical to circular transfer.
    \item It turns out that a Hohmann transfer is the minimum \(\Delta v\) double-impulse maneuver between coplanar circular orbits.
\end{itemize}
\end{block}
\column{0.3\textwidth}
\begin{center}\includegraphics[scale=0.4]{fig_4_9.jpg}\end{center}
\textcolor{blue}{Figure \arabic{section}.9:} Walter Hohmann
\end{columns}
\vspace{-8pt}
\begin{columns}
\column{0.45\textwidth}
\begin{center}\includegraphics[scale=0.8]{fig_4_8.pdf}\end{center}
\vspace{-16pt}
\begin{center}\textcolor{blue}{Figure \arabic{section}.10:} Hohmann transfer\end{center}
\column{0.45\textwidth}
We see that the semimajor axis of the transfer orbit is given by
\[ a_t = \frac{r_1 + r_2}{2} \]
\vfill
\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Plane change maneuvers}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
To change the plane of the orbit, we simply rotate the velocity vector \(\vec{v}\) about the position vector \(\vec{_{}r}\).
\vfill
\begin{center}\includegraphics{fig_4_11.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.11:} Plane change maneuver\end{center}
\vfill
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{columns}
\column{0.45\textwidth}
The velocity change is given by
\[ \Delta v = 2v \sin \frac{\theta}{2} \]
\column{0.45\textwidth}
\begin{center}\includegraphics{fig_4_12.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.12:} Velocity change for plane change maneuver\end{center}
\end{columns}
\vspace{12pt}
For example,
\begin{itemize}
    \item if \(\theta = 60^\circ\), then \(\Delta v = v\).
    \item For a typical low earth orbiting spacecraft, this is approximately 7.5 km/s.
\end{itemize}
\begin{block}{Remark}
If possible, the plane change should be performed at the point of minimum orbital velocity.
\end{block}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The two orbital elements associated with the plane of the orbit are:
\begin{itemize}
    \item the inclination, \( i \)
    \item the right ascension of the ascending node, \(\Omega\)
\end{itemize}
\begin{columns}
\hspace{-20pt}
\column{0.5\textwidth}
\begin{center}\includegraphics[scale=0.9]{fig_2_8.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.13:} Orbital elements\end{center}
\column{0.45\textwidth}
\includegraphics[scale=0.15]{fig_4_14.png}
\textcolor{blue}{Figure \arabic{section}.14:} General plane change maneuver
\end{columns}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
If a pure inclination change is needed, the plane change must occur at the point where the orbit crosses the equatorial plane.
\begin{center}\includegraphics{fig_4_15.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.15:} Plane change maneuver at equatorial crossing\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Combined maneuvers}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\textcolor{blue}{If we wish to change the size, shape and plane of the orbit, combinations of the previous maneuvers are required.} \\
For example, let us consider a transfer between two non-coplanar circular orbits (\(r_2 > r_1\)).
\begin{center}\includegraphics{fig_4_16.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.16:} Combined maneuvers\end{center}
\begin{description}
    \item[Step 1:] Use the first Hohmann transfer at the periapsis of the transfer ellipse.
    \item[Step 2:] Use the second Hohmann transfer and plane change at the apoapsis of the transfer ellipse.
\end{description}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{columns}
\column{0.65\textwidth}
Application for communication satellites
\begin{itemize}
    \item When a satellite is injected into orbit using some launch vehicle, the point of injection is roughly above the launch site.
    \item Since the orbital plane must contain the center of the earth and the insertion point (which is roughly above the launch site), the minimum inclination obtainable is roughly equal to the angle between the launch site and the equatorial plane (also known as the launch site latitude).
    \item Therefore, it is very difficult to launch a satellite directly into an equatorial orbit (unless the launch site is very close to the equator).
\end{itemize}
\column{0.3\textwidth}
\begin{center}\includegraphics{fig_4_17.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.17:} Launch site location\end{center}
\end{columns}
\textcolor{blue}{A common application for the above combined maneuver is transfer into an equatorial orbit.}
\end{frame}
